Trapped rays in cylindrically symmetric media and poles of the analytically continued resolvent

[摘要] The acoustic (stratified) propagator is defined as the positive self-adjoint operator K = -c(2)(y)rho(y)del(z) rho(-1)(y)del(z), on L-2(R-n,(c(2) rho)(-1)d(n)z), where z = (x,y) x is an element of R-k, y is an element of R-m. Bounds on the weighted resolvent of K, defined as < z >(-alpha) (K - zeta)(-1) < z >(-alpha) for alpha > 1/2 and < z > = (1 + \z\(2))(1/2), are obtained at high energies (i.e., in the semi-classical limit Re(zeta) much greater than 0) using the Virial condition c - y.del c greater than or equal to epsilon c > 0. This implies energy decay estimates for trapped waves in fiber optics. With further smoothness assumptions on c and rho the resolvent can be analytically extended onto the second Riemann sheet. The failure of the Virial condition on some radial interval, for m greater than or equal to 2, implies the existence of resonant-traveling modes for the wave equation (partial derivative(t)(2) + K)psi = 0. In particular it is shown that if n > m greater than or equal to 2 then K has resonance states at high energies. No such states exist if m = 1. This is explained using a shape resonance picture. (C) 1997 Academic Press.

[发布日期] 1997-05-15